Strong convergence of random representations of free products of finite groups

We extend the polynomial method of Chen–Garza-Vargas–Tropp–van Handel and Magee–Puder–van Handel for operator-norm bounds in random permutation models to the setting where torsion is present. The main new feature is that asymptotic expansion of traces naturally involves fractional powers of $N$ rather than an ordinary Laurent series. We formulate fractional-power analogues of the method’s key hypotheses and prove they lead to strong convergence. We verify these analogues for free products of finite groups $Γ=G_1*\cdotsG_m$. Concretely, for a uniformly random $ϕ_N\in{\rm hom}(Γ,{\rm Sym}(N))$, set $π_N = {\rm std} \circ ϕ_N$, where ${\rm std}$ denotes the standard $(N-1)$-dimensional representation of ${\rm Sym}(N)$ (the permutation representation with the trivial subrepresentation removed). We deduce strong convergence of $π_N$ to the left regular representation of $Γ$. As applications, we obtain asymptotically sharp spectral gaps for the associated random Schreier graphs, including almost Ramanujan behavior for $C_2C_2C_2$ and an explicit non-Ramanujan limiting spectral radius for $C_2C_3 \cong {\rm PSL}_2({\bf Z})$.

💡 Research Summary

The paper extends the polynomial method, originally developed by Chen, Garza‑Vargas, Tropp, and van Handel and later refined by Magee, Puder, and van Handel, to random permutation models that involve torsion. In the classical setting, the expected trace of a random representation can be expressed as a Laurent series in the dimension N, and the method’s hypotheses (asymptotic expansion, temperedness of leading coefficients, and uniqueness of the trace on the reduced C*‑algebra) guarantee strong convergence of the random representation to the left regular representation. When the underlying group contains elements of finite order, however, the trace expansion acquires fractional powers of N (for example √N in the case of a random involution), and the original hypotheses no longer apply.

The authors formulate fractional‑power analogues of the key hypotheses. Hypothesis A requires that for every group element γ and any precision s≥|γ|, the normalized expected trace E